Extension of Irreducibility results on Generalised Laguerre Polynomials $L_n^{(-1-n-s)}(x)$

Abstract

We consider the irreducibility of Generalised Laguerre Polynomials for negative integral values given by $L_n^{(-1-n-s)}(x)=\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{n-j+s}{n-j}\right)\frac{x^j}{j!}.$ For different values of $s,$ this family gives polynomials which are of great interest. It was proved earlier that for $0\le s\le 60,$ these polynomials are irreducible over $\mathbb{Q}.$ In this paper we improve this result upto $s\le 88.$